Tapered band width dead-end filter



March 12, 1940.

H. A. WHEELER TAPERED BAND WIDTH DEADEND FILTER Filed Oct. 7, 1938 FIG. 3.

INVENTOR HAROLD A.WHEELER BU Y 2 ATTORNEY 2 Sheets-Sheet 1 March 12, 1940.. H A, WHEELER TAPERED BAND WIDTH DEAD-END FILTER Filed Oct. '7, 1958 2 Sheets-Sheet 2 fn fk Frequency Qf) FIG. 5.

L v INVENTOR H OLD A. WHEELER ATTORNEY Patented Mar. l2, 1940 UNITED STATES PATENT ori-lcs 9,192,991 TAPERED BAND WIDTH` DEAD-END FILTER Harold A. WheelerQGrcat Neck, N. Y., assignor to Haleltine Corporation, a corporation of Del- SWC This invention relates generally to coupling systems and particularly to coupling systems including a pair of terminals between which there is l substantial capacitance and across which it is desired to build up a high impedance which is substantially uniform over a wide range of frequencies.

In many coupling arrangements, it is desirable to build up a substantially uniform impedance having a limiting and uniform value over a wide range of frequencies across terminals having susceptance thereacross, that is, it may be desirable to build up a substantially uniform impedance having the highest possible mean value -over a l5 wide range of frequencies across terminals having capacitance in parallel therewith. In other words, it may be desired to maintain uniform at approximately its maximum, the mean value of impedance across the terminals over the wide range of frequencies, or, in other words, to maintain the response of the system substantially uniform and substantially the maximum that can be maintained across the terminals over the wide frequency range. For instance, in the design of vacuum-tube amplifiers to pass a wide range -of frequencies, it is desirable to build up across the inherent capacitance of the -tubes and tube circuits to be coupled the maximum impedance that y can be maintained substantially uniform over the operating frequency range of the amplifier.k The value of the impedance which can be maintained uniform over a given frequency range is limited by the inherent capacitance and inductance of the tube circuits to be coupled. In applicants Patents Nos. 2,167,134; 2,167,135; 2,167,136; and '2,167,137; there are disclosed coupling circuits designed for the above purpose including a deadend filter coupled to the terminals in question and terminated in its image impedance. This 40 mens of terminating well-known types of filters' is based on the concept of designing all the sections of a given filter with the same cutoff frequencies. It is not practicable. with this type of filter termination, to obtain an absolutely uniform response out tothe cutoff frequencies of the system, because attenuation and imageimpedance cannotbe maintainedl exactly uniform as far out as the cutoff frequencies. Y,

It is an object of the present invention, therefore, to provide a coupling system comprising terminals across which there is susceptance wherein a hunting mean value of impedance is maintained across said terminals over a wide frequency range.

It is another object of the invention to provide a coupling system comprising terminals across which there is shunt capacitance wherein a maximum and substantially uniform impedance is maintained across said terminals over a wide'frequency range and in which the response of the system is substantially constant out to the cutoff frequencies of the pass band.

Itis another object of the invention to provide a coupling system of the type described having a maximum impedance over a wide frequency range and suitable for coupling twov successive tubes of a vacuum-tube amplifier.

'I'he present invention relates to filters and in accordance with one embodiment relates to deadend filters in which a uniform image impedance over a wide band is obtained. The method utilizes a filter of tapered band width and involves a departure from confluent filter design in that the cutoff frequencies of the lter as a whole are not critically determined. 'I'he cutoff becomes more gradual and the phase distortion, becomes much less near the cutoff frequencies in a filter of tapered band width. The pass bands of the sections of the filter of the invention increase gradually from section to section near the dead-end termination in order to minimize mismatching of the image impedances at the Junctions of the sections. The filter section at the dead-end termination has a pass band much wider than the use, ful pass band of the entire lter, so that its image impedance is more nearly uniform over the useful frequency band. 'I'he variation of the image impedance of this end section near its own cuto frequencies is of no consequence because that is outside the useful band of the system.

1in-dealing with filters of tapered band width, a new concept of iterative impedance is useful andv is most readily understood by comparison of conventional filters with filters having tapered impedance instead of tapered band width. In the case of filters having a tapered impedance, the iterative impedance varies from section to sec tion, for instance. exponentially, without any change of the frequency scale. 0n the other v hand, a filter of exponentially tapered band width involves a change of the frequency scale from section to section, without any change of the scale of impedance as affecting either the magnitude or the shape of the curve of the iterative impedance. A filter of tapered band width operates as a frequency multiplier in the same manner that va niter .of tapered impedance operates as an im` pedance multiplier. itransformerl.- The iterative impedance characteristics of a-lter oftainstead of a critical point, as inthe case of nontapered filters. The iterative impedance of nlters of taperedband width is complex at all frequencies, although mainly resistive in the pass band and mainly reactive in the attenuation band. The gradualness, of the transition at the cutoif frequency is proportional to the band width ratio from section to section.

In-accordance with a preferred embodiment of the invention, the pass bands of the portions of a multisection dead-end lter are made to increase in a geometric progression toward the dead end oi the filter. The increase of the pass bands widens the image impedancev curves andq the attenuation curves of the successive portions. The useful band is. however, the pass band of that portion of the lter having a minimum pass band. The remaining portions then have substantially no attenuation within the useful band and lss attenuation just outside of the useful band than they would have if they were designed for the same bandwidth, resulting in an overall uniform attenuation characteristic over the useful band, although the attenuation just outside the useful band is not as great as that obtained by the image impedance matching method of terminatingthe dead-end iter. This provides a gradual cutoii' characteristic, which is sometimes desired.

Preferably, the pass band of each portion of the filter is proportioned as an exponential function of the number of the portion, the portions being numbered serially from the dead-end termination of the filter. The exponent of .the function may be any number less than unity. Also, in the preferred embodiment. of the invention, the

above-mentioned portions are filter half-.sections although in other embodiments the filter is tapered by whole-sections.

The novel features believed to be characteristic of the invention are set forth with particularity in the appended claims. 'I'he invention itself, however, both as to its organization and method of operation, together with other and further objects and advantages thereof, will ibest be understood by reference to the specification taken in connection with the accompanying drawings. in which Fig. 1 is a circuit diagram of a low-pass lter of exponential taper; Fig. 2 is a graph representing the impedance characteristics of the of the filter of Fig. l; Fig. 3 is a circuit diagram of a filter which is tapered from section to section; Fig. 4 is a circuit diagram of a lter which is tapered by half-sections; Fig. 5 is a graph illustrating certain of the operating characteristics of the circuit of Fig. 4; Fig. 6 is a circuit diagram of a band-pass filter tapered by sections; Fig. 7 shows the transfer impedance characteristics of the circuit of Fig. 6; Fig. 8 is a circuit diagram of a band-pass amplier embodying a practical arcano:

the circuit values vary exponentially along the filter instead of being the same in each section.'

The pass bands of the sections increase through a large number of sections and the filter is terminated by a resistor R0. 'I'here is lexact matching of the image impedance at zero frequency at all Junctions and at the terminal resistor Re. Therefore, the ratio of capacitance to inductance is the same in all half-sections, while the product of capacitance and inductance in any section is inversely proportional to the pass band of that section. The half-sections may be numbered serially from the terminal resistor R0, the nth half-section being at the end of the filter remote from Ro. The band width progressively increases from half-section to half-section toward the far end in the inverse order of the serial number. The pass bands of the iilter portions and the resistance termination are so proportioned with regard to the capacitance across its input terminals that the response of the system is substantially uniform over the useful pass band ol the system and is substantially the maximum that can be maintained by the input terminals over the range.

The filter of Fig. 1 thus comprises a pair of terminals ,I, 2, across which is coupled `condenser cCn and the impedance across vwhich is Z, the admittance looking into the n sections of the filter coupled tol terminals l, 2 being designated Yu.

The filter thus comprises only a part (Cn) of the capacitance across terminals I, 2 as a terminal mid-element of the lter. Coupled to the pair of terminals I, 2A is a series of filter portions tera is the ratio of the-pass bands of consecutive filter half-sections.

In terms of the cutoii frequency fn of the nth half-section, that of the first half-section is:

The nth half-section includes the elements Cn and Ln and in terms of these elements, the circuit values of the first section are:

and the circuit values of the kth section are:

The total capacitance and inductance of a large number of half-sections approach the limiting values:

The effect of an iniinite number of filter portions is closely approximated by a few filter portions at the end remote from the dead end, with the remainder of the total inductance and capacitance inserted in single lumps.

Fig. 2 shows the impedance and admittance characteristic of the filter of Fig. l, assuming an infinite number of sections. The dotted curves are ideal curves based on exact matching and the full-line curves represent corresponding characteristics of the tapered lter of the invention.

arcano:

Curve Y shows the admittance across terminals l, 2. Curve Ya shows the vadmittance characteristic looking through n half-sections, curve Zn-i the relative impedance looking through n-l halfsections, etc. All the curves have the same shape but the frequency pass bands are multiplied by the/constante from curve to curve. `Adjacent curves are plotted in termsof inverse quantities, admittanceand impedance to glve a family of similar curves, because each half-section inverts the impedance characteristic in addition to changing the scale of frequency. This inversion is carried over from the inversion ofthe image impedance by a half-section.

Y The curves of Fig. 2 are drawn` approximately to show the general shape. 'I'hey are susceptible of computation in terms of an infinite series, or as the limit approached by increasing the number of half-sections. They represent a new impedance function which is a continuous function of frequency approximating the usual discontinuous'image impedance. Since it is continuous, it can be realized very closely in a finite network.

The iterative impedance of. the filter with exponentially tapered band width depends not only on the image impedance of the individual sections, but also on their phase delay and attenuation. Therefore, different sections. having the same image impedance do not present the same iterative impedance with tapered band width,

although the difference may be small. Since this iterative impedance approximates the nonuniform image impedance of a confluent filter, a filter with tapered band width may be used instead of an m-derived filter to match a nonuniform image impedance of a confluent lter with a constant` resistance. The approximation is closer the less the rate of taper, involving a filter of more sections. I

A tapered band width filter is especially useful as a dead-end filter to securemaximum uniform impedance Z0 across a given value of. capacitance C0 over a given band width of Aw. The iigure of merit for a network for this purpose has the upper limiting value:

which is approached by a conventional filter network of many sections. 'I'his upper limit may be approached with fewer sections by the use of a tapered band width filter.

The synthesis of this uniform impedance 4involves first setting up a network in which a constant-k mid-shunt image impedance is approximately realized and then connecting in parallel a constant-k: mid-shunt arm. This procedurel is fully described and claimed in the above-mentioned patents. Fig. 1 illustrates a variation of this method as applied to a filter of exponentially tapered band widths. 'I'he capacitance aCn`.- to be connected across Cn is the mid-shunt arm of the (n+1)th half-section in the series. mid-shunt armv aCn is related in the same manner as that in which elements of like kind are proportioned in the successive filter portions of thetapered band width filter to which aCn is coupled. The resulting impedance Z is'built up across the total capacitance:

This has a magnitude approximately constant That is, the

, 3 and'equalgto Re nearly up to the cutoff frequency:

The magnitude of the impedance decreases gradually in the cutoff region, as shown by the characteristic of the curve Y=1/Z in Fig. 2.

Since the results obtained by this method are only an approximation of theideal. the relations are not critical.

They may be modified by experiment if need be. Therefore, approximate simple rules of design are useful. Fig. 3 is a circuit diagram of a tapered band width filter tapered in terms -of full-series and full-shunt filter arms, instead of half-sections, as was Vdone in the case of Fig. 1 and vcomprising onlytwo full sections. The tapering factor a is applied in Aprogressively increasing powers to the successive may not be the optimum compromise. Also,

.there'is the question of what the rate of taper should be for a filter of a given number of sections or half-sections. Fig. 4 is a circuit diagram of a tapered dead-end filter using another rule of taper for developing a uniform impedance over the useful frequency band across the terminal capacitance:

Co=C4+Cs (10) This filter compises four half-sections comprising mid-series inductance arms L1, L2, etc., and mid-shunt capacitance arms C1. Cz, etc.. and midshunt arm C5 of a fifth half-section, numbered serially from the dead-end termination comprising resistor Re. The `cutoff lfrequencies of the individual half-sections may likewise be deter- -mined as a similar function of their serial numbers to obtain an optimum relation between the .cutoff frequencies.

The curves of Fig. 5 indicate the mld-shunt image admittance or the mid-series impedance characteristics of certain half-sections in the tapered filterof Fig. 4. curve o representing the uniform impedance or admittance of the terminal resistor, curve k representing the characwith tapered band width, the exponential taper teristic of any half-section of serial number lc.

and curve n the corresponding characteristic of the terminalhalf-section. It is desirable to distribute the mismatching of. the image impedances equallyamong the several junctions of the j filter so as to minimize its effect. This means that the intermediate curves k should fall between the extreme curves in geometric progression on the scale of ordinates.` This is approximately realized over the majority of the useful band if the curves are luniformly spaced at the lower frequencies. e f

In order to secure this spacing. the passbands of the filter sections should be tapered in the inverse ratio of the square root of their serial In the low-pass example of Fig. 4, the circuit values for this relationship are:

Aand 4. Fig. 6 is a circuit diagram of a bandpass example which cannot be divided into 'halfsections by the usual rules. The tapering of the filter of Fig. 6 is by whole-sections, of which two are shown, the series inductance of each section being designated by the subscript s in addition to the subscript relating to the serial number of its section and the parallel-connected inductance being correspondingly designated by th subscript 1J. The third portion of the lter is a constant-k mid-shunt arm comprising parallel-connected inductance La and capacitance C3. This lter is preferably designed by the square-root rule of tapering and then presents uniform impedance across the terminal capacil tance AC3 over the useful band width. Also, it presents uniform transfer impedance between the two pairs of terminals designated.

The curves of Fig. '7 show the relative transfer impedance of a four-terminal network having the circuit of Fig. 6. 'I'he terminals Il, 2 and 3, 4 of the ilter are indicated in Fig. 6 and are separated by one section of the filter at the end remote from the dead end. 1 The curves are plotted in terms of the frequency departure from mid-band relative to the mean frequency of the band:

in which Ca and Cb are; respectivelythe total capacitance across the input terminals I, 2 and across the output terminals 3, I, and Zo is the mid-band transfer impedance.

In Fig. 7, curve a is the idealized curve for a' large number of sections; curve b is computed vfor the particular lter circuit of Fig. 6 in which the cutoff frequencies, in terms of departure from mid-band, vare in the proportion:

Curve c represents the characteristic for the lter of Fig. 6 tapered in accordance with the square-root rule:

1 =2.76; j-r2=1.94; aus (16) fo n vf Curve d represents the lter characteristic for untapered band width. The square-root rule gives a good practical compromise.

The terminals of the dead-end iilters above described have been indicated. It will be understood, however, that the circuits may be used having a singleterminal circuit or having a plurality of terminal circuits in the manner I described in the above-mentioned patents.

Fig. 8 is a circuit diagram of a band-pass ampliiier using' a practical circuit equivalent to the filter of Fig. 6 for coupling the output circuit of a vacuum tube I0 to the input circuit of a vacuum tube il. This lter comprises three resonant circuits La, Cs; Lb', Lb", Cb; and La, Cs. Condensers C and Cb have been shown by dotted lines to indicate that they may be comprised in whole or part of the inherent capacitance of the circuit in which they are connected. Since no coupling is desired between tuned circuit L. C and tuned circuit Le, Cc, the circuit Lb. Ls, Cb has its inductance split into two parts Lb' and Lb", coupled respectively with L. and Lc of the iirst and third circuits. If parts Lb and Lb" were equal, the coefficient of coupling vIt' between inductances La and Lb' would .be much less than the coefficient of coupling 1c" between inductances Ln" and Le. These coeicients are equalized by making Lb less than In". Thecoil arrangement shown has each pair of coupled coils on one of two parallel coil forms I2 and i3. Zero coupling between inductances La and Le is procured by separating the coil forms a suitable distance andso disposing them that their line of'centers and their axes make an angle of about 55 degrees. Coils Ls and Le" are at least approximately uncoupled so that they can be moved relative to coils L. and Le, respectively, for adjusting the coupling between successive circuits without lappreciable change of the total inductance of the circuit Lb', Lb", Cn. 'Ihe dead-end circuit may have any convenient value of impedance so that Le may be identical to Lb'. The shunt condenser Cc and resistor Ro' are proportioned accordingly. All three circuits are individually resonant at or near the mldband frequency of the system.

In the design of the circuit of Fig. 8 to meet the critical conditions which give curve bof lFig.'7, the following formulae-are followed; theyvare applicable when the band width is much less A'than the mean frequency of the band. First, each capacitance C and Cb is determined; then each inductance is computedvto resonate each circuit i 1 Y L.=L1,f; C.=Lqwma (19) The coupling and damping determine the band width. By reference to curve b of Fig. '1, the value of we or fn is chosen, which is about half the band with. From thisv ,value the coupling an damping are computed:

The resulting transfer impedance at mid-band is:

If the band width is comparable withthe mean formulae fail, the circuit is designed from filter i formulae starting with Fig. 6. The geometric mean frequency is the same in all sections and the image impedance is matched at each Junction at that frequency. Any desired rule of tapering may be employed to determine the relative band widths of the filter section. The dead-end circuit Cc, L.; may be utilized also to couple the signal to another tube in an auxiliary channel of the system.

Fig. 9 is a circuit diagram of a simplied arrangement equivalent to the circuit of Fig. 8. The circuit of Fig. 9 requires less. coupling between the windings of the system and requires fewer coils, but has less freedom in adjusting the coupling and `adjusting the individual circuits. Elements which are identical in the two gures have been given similar reference numerals. The following formulae are applicable to the circuit of Fig. 9 if the band width is much less than the mean frequency of the system. The formulae for La and Lb are the same as Equation 1'7 above:

The formulae for Gc and Zo are the same as Equations 21 and 22 above.

It will be understood that the square-root rule of tapering generally gives an approximation for the critical conditions of a uniform impedance A over a wide frequency band. If a wider band is desired at the expense of uniformity of impedance, this rule may be compromised by utilizing the cube-root or even the fourth-root in the taper formulae, the results then being more like the untapered filter` 'I'hat is to say, instead of proportioning the filter portions so that their band widths vary inversely to the square-root of the serial number of filter portions from the dead end of the filter, the band widths of the filter portions may be proportioned to vary inversely in accordance with a greater root of the serial number of filter portions.

While there have been described what are at present considered to be the preferred embodiments of this invention, it will be obvious to those skilled in the art that various changes and modifications may be made therein without departing from the invention, and it is. therefore, aimed in the appended claims to cover all such changes and modifications as fall within the true spirit and scope of the invention.

.What is claimed is:

1. A signal-translating system for operation over a range of frequencies comprising at least one pair of terminals between which there is substantial reactance tending to render the response of said system nonuniform over said range, a multisection dead-end filter coupled to said pair of terminals, said filter comprising only part of said reactance as a terminal, mid-element of said filter, said filter comprising portions the band widths of which increase progressively toward said dead end, a resistance termination for said filter at said dead end matching the image impedance thereof over the useful pass band, the pass bands of said filter portions and said resistance termination being so proportioned with respect to said reactance that the response of said.

system is substantially uniform over the useful pass band of said system and is substantially the maximum that can be maintained between said pair of terminals over said range.

2. A signal-translating system for operation over a range of frequencies comprising at least oneV pair of terminals between which there is substantial capacitance tending to render the re sponse of said system nonuniform over said range, a multisection dead-end filter coupled to said pair of terminals, said filter comprising only part of said capacitance as a terminal mid-shunt element of said filter, said iter comprising portions the band widths of which increase progressively toward said dead end, a resistance termination for said filter at said dead end matching the image impedance thereof over the useful pass band, the band widths of said lter portions and said resistance termination being so proportioned with respect to said capacitance that the response of said system is substantially uniform over the useful pass band of said system and is substantially the maximum that can be maintained between said pair of terminals over said range.

3. A signal-translating system for operation over a range of frequencies comprising at least one pair of terminals between which there is substantial reactance tending to render the response of said system nonuniform over said range, a multisection dead-end filter coupled to said pair of terminals, said filter comprising only part of said reactance as a terminal mid-element of said filter, said filter comprising a series of halfsections the band widths of which increase toward said dead end, a resistance termination for said filter at said dead end matching the image impedance of said filter over the useful band width, the band widths of said half-sections and said resistance termination being so proportioned with respect to said reactance that the response of said system is substantially uniform over the useful pass'band of said system and is substantially the maximum that can be maintained between said pair of terminals over said range.

4. A signal-translating system for operation over a range of frequencies comprising at least one pair of terminals between which there is substantial reactance tending to render the response of said system nonuniform over said range, a multisection dead-end filter coupled to said pair of terminals, said filter comprising only part of said reactance as a terminal mid-element of said filter, said filter comprising filter whole-sections the band widths of which increase toward said dead end, a resistance termination for said filter at said dead end matching the image impedence of said lter over the useful band width, the band widths of said lter whole-sections and said resistance termination being so proportioned with vrespect to said reactancethat the response of said system is substantially uniform over the useful pass band of said system and is substantially the maximum that can be maintained between said pair of terminals over said range.

5. A signal-translating system for operation over a range of frequencies comprising at least one pair of terminals between which there is substantlalreactance tending to renderthe response of said system nonuniform over said range, a multisection dead-end filter coupled to said pair of terminals, said filter comprising only part of said reactance as a terminal mid-element of said filter, said lter comprising portions the band widths of which vary exponentially toward said vdead end, said mid-element bearing an exponential relation to elements of-like kind in said portions similar to that between said elements of like kind, and a resistance termination for said filter at said dead end matching the image impedance of saidfllter over the useful band width, whereby the response of said system is substantially uniform over the useful pass band of said system and is substantially the maximum that can be maintained between said pair of terminals over said range.

6. A signal-translating system for operation over a range of frequencies comprising at least one pair of terminals between which there is substantial reaetance tending to render the response of said system nonunlform over said range, a multisection dead-end filter coupled to said pair of terminals, said filter comprising only part of said reaetance as a terminal mid-element of said filter, said filter comprising portions the band widths of which vary inversely in accordance with the square-root of the serial numbers of the portions counted from said dead end, said mid-element bearing a relation to elements of like kind in said nlter portions similar to that between said elements of like kind, and a resistance termination matching the image impedance of said filter at said dead end, whereby the response of said system is substantially uniform over the useful pass band ofsaid system and is substantially the maximum that can be maintained between said pair of terminals over said range.

7. A signal-translating system for operation over a range of frequencies comprising at least one pair of terminals between which there is substantial reaetance tending to render the response of said system nonuniform over said range,v a multisection dead-end lter coupled to saidl pair of terminals, said filter comprising only part of said reaetance as a terminal mid-element of said nlter, said filter comprising portions, the band widths of which vary inversely as a root of an order higher than the square-root of the serial numbers of the portions counted from said dead end, said mid-element bearing a relation to elements of like kind in said filter portions similar to that between said elements of like kind, a ,resistance termination matching the image impedance of said iilter at said dead end, wherebycomprising portions including a mid-series arm having inductance and a mid-shunt arm comprising a parallel resonant circuit, the band widths of said portions increasing toward said dead end, a resistance termination for said lter at said dead end lmatching the image impedance of said filter over the useful band width, the band widths of said filter portions and said resistance termination being so proportioned with respect to said reaetance that the response of said system issubstantially uniform over the useful pass band of said system and is -substantially the maximum that can be maintained between said pair of terminals over said range.

9. A low-pass signal-,translating system for operation over a range of frequencies comprising l at least one pair of terminals between which there is substantial reaetance tending to limit the response of said system over said range, a multisection low-pass dead-end filter coupled to said pair of terminals, said filter comprising only part of said reaetance as a terminal mid-element of said -filter, said filter comprising portions including series inductance and shunt capacitance, the band widths of said portions increasing toward said dead end, a resistance termination for said lter at said dead end matching the image impedance of said filter over the useful band width, the band widths of said filter portions and said resistance termination being so proportioned with respect to said reactance that the response of said system is substantially uniform over the useful pass band of said system and is substantially the maximum that can be maintained between said pair of terminals over said range.

' HAROLD A. WHEELER. 

